Question

The value of \[ \sin(x+y)\sec x\sec y \] is

• A. \(\cos x\cos y\)
• B. \(\tan x-\tan y\)
• C. \(\cos x+\cos y\)
• D. \(\tan x+\tan y\)

Hint:

To simplify trigonometric products involving \(\sec x\) and \(\sec y\), first expand compound angles and then convert \(\sec\) into reciprocal cosine.

Answer 1

The Correct Option is D

Step 1: Expand \(\sin(x+y)\).
Using the identity, \[ \sin(x+y)=\sin x\cos y+\cos x\sin y \]

Step 2: Substitute in the given expression.
\[ \sin(x+y)\sec x\sec y \] \[ =(\sin x\cos y+\cos x\sin y)\sec x\sec y \]

Step 3: Use \(\sec x=\frac{1}{\cos x}\).
\[ =(\sin x\cos y+\cos x\sin y)\frac{1}{\cos x}\frac{1}{\cos y} \] \[ =\frac{\sin x\cos y}{\cos x\cos y} + \frac{\cos x\sin y}{\cos x\cos y} \] \[ =\frac{\sin x}{\cos x} + \frac{\sin y}{\cos y} \] \[ =\tan x+\tan y \]

Step 4: Final conclusion.
Therefore, \[ \boxed{\tan x+\tan y} \]